Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
tg+cos2a-sin2a/(1+sina*cosa)*cosa
¿Cómo vas a descomponer esta tg+cos2a-sin2a/(1+sina*cosa)*cosa expresión en fracciones?
Solución
Ha introducido
[src]
sin(2*a)
tan(x) + cos(2*a) - -----------------*cos(a)
1 + sin(a)*cos(a)
$$- \frac{\sin{\left(2 a \right)}}{\sin{\left(a \right)} \cos{\left(a \right)} + 1} \cos{\left(a \right)} + \left(\cos{\left(2 a \right)} + \tan{\left(x \right)}\right)$$
tan(x) + cos(2*a) - sin(2*a)/(1 + sin(a)*cos(a))*cos(a)
Simplificación general
[src]
/ sin(2*a)\
|1 + --------|*(cos(2*a) + tan(x)) - cos(a)*sin(2*a)
\ 2 /
----------------------------------------------------
sin(2*a)
1 + --------
2
$$\frac{\left(\frac{\sin{\left(2 a \right)}}{2} + 1\right) \left(\cos{\left(2 a \right)} + \tan{\left(x \right)}\right) - \sin{\left(2 a \right)} \cos{\left(a \right)}}{\frac{\sin{\left(2 a \right)}}{2} + 1}$$
((1 + sin(2*a)/2)*(cos(2*a) + tan(x)) - cos(a)*sin(2*a))/(1 + sin(2*a)/2)
Unión de expresiones racionales
[src]
(1 + cos(a)*sin(a))*(cos(2*a) + tan(x)) - cos(a)*sin(2*a)
---------------------------------------------------------
1 + cos(a)*sin(a)
$$\frac{\left(\sin{\left(a \right)} \cos{\left(a \right)} + 1\right) \left(\cos{\left(2 a \right)} + \tan{\left(x \right)}\right) - \sin{\left(2 a \right)} \cos{\left(a \right)}}{\sin{\left(a \right)} \cos{\left(a \right)} + 1}$$
((1 + cos(a)*sin(a))*(cos(2*a) + tan(x)) - cos(a)*sin(2*a))/(1 + cos(a)*sin(a))
Potencias
[src]
cos(a)*sin(2*a) - ----------------- + cos(2*a) + tan(x) 1 + cos(a)*sin(a)
$$\cos{\left(2 a \right)} + \tan{\left(x \right)} - \frac{\sin{\left(2 a \right)} \cos{\left(a \right)}}{\sin{\left(a \right)} \cos{\left(a \right)} + 1}$$
/ I*a -I*a\
|e e | / -2*I*a 2*I*a\
-2*I*a 2*I*a / I*x -I*x\ I*|---- + -----|*\- e + e /
e e I*\- e + e / \ 2 2 /
------- + ------ + ------------------ + -----------------------------------------
2 2 I*x -I*x / / I*a -I*a\ \
e + e | |e e | / -I*a I*a\|
| I*|---- + -----|*\- e + e /|
| \ 2 2 / |
2*|1 - ---------------------------------|
\ 2 /
$$\frac{i \left(- e^{i x} + e^{- i x}\right)}{e^{i x} + e^{- i x}} + \frac{e^{2 i a}}{2} + \frac{e^{- 2 i a}}{2} + \frac{i \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right) \left(e^{2 i a} - e^{- 2 i a}\right)}{2 \left(- \frac{i \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right) \left(e^{i a} - e^{- i a}\right)}{2} + 1\right)}$$
exp(-2*i*a)/2 + exp(2*i*a)/2 + i*(-exp(i*x) + exp(-i*x))/(exp(i*x) + exp(-i*x)) + i*(exp(i*a)/2 + exp(-i*a)/2)*(-exp(-2*i*a) + exp(2*i*a))/(2*(1 - i*(exp(i*a)/2 + exp(-i*a)/2)*(-exp(-i*a) + exp(i*a))/2))
Denominador común
[src]
cos(a)*sin(2*a) - ----------------- + cos(2*a) + tan(x) 1 + cos(a)*sin(a)
$$\cos{\left(2 a \right)} + \tan{\left(x \right)} - \frac{\sin{\left(2 a \right)} \cos{\left(a \right)}}{\sin{\left(a \right)} \cos{\left(a \right)} + 1}$$
-cos(a)*sin(2*a)/(1 + cos(a)*sin(a)) + cos(2*a) + tan(x)
Compilar la expresión
[src]
cos(a)*sin(2*a) - ----------------- + cos(2*a) + tan(x) 1 + cos(a)*sin(a)
$$\cos{\left(2 a \right)} + \tan{\left(x \right)} - \frac{\sin{\left(2 a \right)} \cos{\left(a \right)}}{\sin{\left(a \right)} \cos{\left(a \right)} + 1}$$
-cos(a)*sin(2*a)/(1 + cos(a)*sin(a)) + cos(2*a) + tan(x)
Denominador racional
[src]
(1 + cos(a)*sin(a))*(cos(2*a) + tan(x)) - cos(a)*sin(2*a)
---------------------------------------------------------
1 + cos(a)*sin(a)
$$\frac{\left(\sin{\left(a \right)} \cos{\left(a \right)} + 1\right) \left(\cos{\left(2 a \right)} + \tan{\left(x \right)}\right) - \sin{\left(2 a \right)} \cos{\left(a \right)}}{\sin{\left(a \right)} \cos{\left(a \right)} + 1}$$
((1 + cos(a)*sin(a))*(cos(2*a) + tan(x)) - cos(a)*sin(2*a))/(1 + cos(a)*sin(a))
Respuesta numérica
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-cos(a)*sin(2*a)/(1.0 + cos(a)*sin(a)) + cos(2*a) + tan(x)
-cos(a)*sin(2*a)/(1.0 + cos(a)*sin(a)) + cos(2*a) + tan(x)
Combinatoria
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-cos(a)*sin(2*a) + cos(a)*cos(2*a)*sin(a) + cos(a)*sin(a)*tan(x) + cos(2*a) + tan(x)
------------------------------------------------------------------------------------
1 + cos(a)*sin(a)
$$\frac{\sin{\left(a \right)} \cos{\left(a \right)} \cos{\left(2 a \right)} + \sin{\left(a \right)} \cos{\left(a \right)} \tan{\left(x \right)} - \sin{\left(2 a \right)} \cos{\left(a \right)} + \cos{\left(2 a \right)} + \tan{\left(x \right)}}{\sin{\left(a \right)} \cos{\left(a \right)} + 1}$$
(-cos(a)*sin(2*a) + cos(a)*cos(2*a)*sin(a) + cos(a)*sin(a)*tan(x) + cos(2*a) + tan(x))/(1 + cos(a)*sin(a))
Parte trigonométrica
[src]
/pi \
csc|-- - x|
1 \2 / 1
------------- + ----------- - ---------------------------------------------
/pi \ csc(x) / 1 \ /pi \
csc|-- - 2*a| |1 + ------------------|*csc(2*a)*csc|-- - a|
\2 / | /pi \| \2 /
| csc(a)*csc|-- - a||
\ \2 //
$$\frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} - \frac{1}{\left(1 + \frac{1}{\csc{\left(a \right)} \csc{\left(- a + \frac{\pi}{2} \right)}}\right) \csc{\left(2 a \right)} \csc{\left(- a + \frac{\pi}{2} \right)}}$$
2*cos(a)*sin(2*a)
- ----------------- + cos(2*a) + tan(x)
2 + sin(2*a)
$$\cos{\left(2 a \right)} + \tan{\left(x \right)} - \frac{2 \sin{\left(2 a \right)} \cos{\left(a \right)}}{\sin{\left(2 a \right)} + 2}$$
2
sin(a) + sin(3*a) 2*sin (x) /pi \
- ----------------- + --------- + sin|-- + 2*a|
2 + sin(2*a) sin(2*x) \2 /
$$- \frac{\sin{\left(a \right)} + \sin{\left(3 a \right)}}{\sin{\left(2 a \right)} + 2} + \frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + \sin{\left(2 a + \frac{\pi}{2} \right)}$$
1 1
----------- + -------------
/ pi\ / pi\
sec|a - --| sec|3*a - --|
1 sec(x) \ 2 / \ 2 /
-------- + ----------- - ---------------------------
sec(2*a) / pi\ 1
sec|x - --| 2 + -------------
\ 2 / / pi\
sec|2*a - --|
\ 2 /
$$\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(2 a \right)}} - \frac{\frac{1}{\sec{\left(3 a - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}}{2 + \frac{1}{\sec{\left(2 a - \frac{\pi}{2} \right)}}}$$
/a\ /3*a\
2*tan|-| 2*tan|---|
\2/ \ 2 /
----------- + -------------
2/a\ 2/3*a\
2 1 + tan |-| 1 + tan |---|
1 - tan (a) \2/ \ 2 /
----------- - --------------------------- + tan(x)
2 2*tan(a)
1 + tan (a) 2 + -----------
2
1 + tan (a)
$$\frac{1 - \tan^{2}{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} + \tan{\left(x \right)} - \frac{\frac{2 \tan{\left(\frac{3 a}{2} \right)}}{\tan^{2}{\left(\frac{3 a}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}}{2 + \frac{2 \tan{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1}}$$
/ 2/a\\
2 2*|-1 + cot |-||*cot(a)
1 -1 + cot (a) \ \2//
------ + ------------ - ---------------------------------------------------------
cot(x) 2 / / 2/a\\ /a\\
1 + cot (a) | 2*|-1 + cot |-||*cot|-||
/ 2 \ / 2/a\\ | \ \2// \2/|
\1 + cot (a)/*|1 + cot |-||*|1 + -----------------------|
\ \2// | 2 |
| / 2/a\\ |
| |1 + cot |-|| |
\ \ \2// /
$$\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} + \frac{1}{\cot{\left(x \right)}} - \frac{2 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right) \cot{\left(a \right)}}{\left(\frac{2 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right) \cot{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\cot^{2}{\left(a \right)} + 1\right)}$$
sin(a) + sin(3*a)
- ----------------- + cos(2*a) + tan(x)
2 + sin(2*a)
$$- \frac{\sin{\left(a \right)} + \sin{\left(3 a \right)}}{\sin{\left(2 a \right)} + 2} + \cos{\left(2 a \right)} + \tan{\left(x \right)}$$
/ 2/a\\
2 2*|1 - tan |-||*tan(a)
1 - tan (a) \ \2//
----------- - -------------------------------------------------------- + tan(x)
2 / / 2/a\\ /a\\
1 + tan (a) | 2*|1 - tan |-||*tan|-||
/ 2 \ / 2/a\\ | \ \2// \2/|
\1 + tan (a)/*|1 + tan |-||*|1 + ----------------------|
\ \2// | 2 |
| / 2/a\\ |
| |1 + tan |-|| |
\ \ \2// /
$$- \frac{2 \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right) \tan{\left(a \right)}}{\left(\frac{2 \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right) \tan{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} + 1\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\tan^{2}{\left(a \right)} + 1\right)} + \frac{1 - \tan^{2}{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} + \tan{\left(x \right)}$$
2 2*sin (x) cos(a)*sin(2*a) --------- - ----------------- + cos(2*a) sin(2*x) 1 + cos(a)*sin(a)
$$\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + \cos{\left(2 a \right)} - \frac{\sin{\left(2 a \right)} \cos{\left(a \right)}}{\sin{\left(a \right)} \cos{\left(a \right)} + 1}$$
sin(x) cos(a)*sin(2*a) ------ - ----------------- + cos(2*a) cos(x) 1 + cos(a)*sin(a)
$$\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + \cos{\left(2 a \right)} - \frac{\sin{\left(2 a \right)} \cos{\left(a \right)}}{\sin{\left(a \right)} \cos{\left(a \right)} + 1}$$
/ pi\ / pi\ / pi\
cos|x - --| cos|a - --| + cos|3*a - --|
\ 2 / \ 2 / \ 2 /
----------- - --------------------------- + cos(2*a)
cos(x) / pi\
2 + cos|2*a - --|
\ 2 /
$$- \frac{\cos{\left(a - \frac{\pi}{2} \right)} + \cos{\left(3 a - \frac{\pi}{2} \right)}}{\cos{\left(2 a - \frac{\pi}{2} \right)} + 2} + \cos{\left(2 a \right)} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}$$
/ pi\
2 sin(2*a)*sin|a + --|
2*sin (x) \ 2 / /pi \
--------- - ---------------------- + sin|-- + 2*a|
sin(2*x) / pi\ \2 /
1 + sin(a)*sin|a + --|
\ 2 /
$$\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + \sin{\left(2 a + \frac{\pi}{2} \right)} - \frac{\sin{\left(2 a \right)} \sin{\left(a + \frac{\pi}{2} \right)}}{\sin{\left(a \right)} \sin{\left(a + \frac{\pi}{2} \right)} + 1}$$
/ pi\ / pi\
cos|x - --| cos(a)*cos|2*a - --|
\ 2 / \ 2 /
----------- - ---------------------- + cos(2*a)
cos(x) / pi\
1 + cos(a)*cos|a - --|
\ 2 /
$$\cos{\left(2 a \right)} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}} - \frac{\cos{\left(a \right)} \cos{\left(2 a - \frac{\pi}{2} \right)}}{\cos{\left(a \right)} \cos{\left(a - \frac{\pi}{2} \right)} + 1}$$
/a\ /3*a\
2*cot|-| 2*cot|---|
\2/ \ 2 /
----------- + -------------
2/a\ 2/3*a\
2 1 + cot |-| 1 + cot |---|
1 -1 + cot (a) \2/ \ 2 /
------ + ------------ - ---------------------------
cot(x) 2 2*cot(a)
1 + cot (a) 2 + -----------
2
1 + cot (a)
$$\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} + \frac{1}{\cot{\left(x \right)}} - \frac{\frac{2 \cot{\left(\frac{3 a}{2} \right)}}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}}{2 + \frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1}}$$
1 sec(x) 1
-------- + ------ - -----------------------------------
sec(2*a) csc(x) / 1 \
|1 + -------------|*csc(2*a)*sec(a)
\ csc(a)*sec(a)/
$$\frac{1}{\sec{\left(2 a \right)}} + \frac{\sec{\left(x \right)}}{\csc{\left(x \right)}} - \frac{1}{\left(1 + \frac{1}{\csc{\left(a \right)} \sec{\left(a \right)}}\right) \csc{\left(2 a \right)} \sec{\left(a \right)}}$$
1 sec(x) 1
-------- + ----------- - ---------------------------------------------
sec(2*a) / pi\ / 1 \ / pi\
sec|x - --| |1 + ------------------|*sec(a)*sec|2*a - --|
\ 2 / | / pi\| \ 2 /
| sec(a)*sec|a - --||
\ \ 2 //
$$\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(2 a \right)}} - \frac{1}{\left(1 + \frac{1}{\sec{\left(a \right)} \sec{\left(a - \frac{\pi}{2} \right)}}\right) \sec{\left(a \right)} \sec{\left(2 a - \frac{\pi}{2} \right)}}$$
/pi \ 1 1
csc|-- - x| ------ + --------
1 \2 / csc(a) csc(3*a)
------------- + ----------- - -----------------
/pi \ csc(x) 1
csc|-- - 2*a| 2 + --------
\2 / csc(2*a)
$$\frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} - \frac{\frac{1}{\csc{\left(3 a \right)}} + \frac{1}{\csc{\left(a \right)}}}{2 + \frac{1}{\csc{\left(2 a \right)}}}$$
1/csc(pi/2 - 2*a) + csc(pi/2 - x)/csc(x) - (1/csc(a) + 1/csc(3*a))/(2 + 1/csc(2*a))